Combined WENO schemes for increasing the accuracy of the numerical solution of conservation laws

TL;DR

This paper introduces a new WENO scheme that leverages all sub-stencils to enhance accuracy in solving conservation laws, maintaining non-oscillatory properties and tested on wave and Euler equations.

Contribution

A novel WENO method that utilizes all sub-stencils for higher accuracy while preserving the scheme's non-oscillatory nature.

Findings

Increased accuracy in numerical solutions.

Maintains non-oscillatory property near discontinuities.

Effective in linear and Euler equations test cases.

Abstract

In this article, we introduce a new method which allows utilizing all the available sub-stencils of a WENO scheme to increase the accuracy of the numerical solution of conservation laws while preserving the non-oscillatory property of the scheme. In this method, near a discontinuity, if there is a smooth sub-stencil with higher-order of accuracy, it is used in the reconstruction procedure. Furthermore, in smooth regions, all the sub-stencils of the same order of accuracy form the stencil with the highest order of accuracy as the conventional WENO scheme. The presented method is assessed using several test cases of the linear wave equation and one- and two-dimensional Euler's equations of gas dynamics. In addition to the original weights of WENO schemes, the WENO-Z approach is used. The results show that the new method increases the accuracy of the results while properly maintaining the ENO property.